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Consider a matrix polynomial $P \left( \lambda \right)= A_0 + \lambda A_1 +\ldots + \lambda^d A_d$, with $A_0,\ldots, A_d$ complex (or real) matrices witha certain structure. In this paper we discuss an iterative method tonumerically approximate the closest structured singular matrix polynomial$\widetilde P\left( \lambda \right)$, using the distance induced by theFrobenius norm. An important peculiarity of the approach we propose is thepossibility to include different types of structural constraints. The methodalso allows us to limit the perturbations to just a few matrices and also toinclude additional structures, such as the preservation of the sparsity patternof one or more matrices $A_i$, and also collective-like properties, like apalindromic structure. The iterative method is based on the numericalintegration of the gradient system associated with a suitable functional whichquantifies the distance to singularity of a matrix polynomial.

Approximating the closest structured singular matrix polynomial